Identifier
Values
[1] => ([],1) => ([],1) => ([],1) => 1
[1,1] => ([(0,1)],2) => ([],1) => ([],1) => 1
[2] => ([],2) => ([],2) => ([(0,1)],2) => 1
[1,2] => ([(1,2)],3) => ([],2) => ([(0,1)],2) => 1
[2,1] => ([(0,2),(1,2)],3) => ([],1) => ([],1) => 1
[3] => ([],3) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 2
[1,3] => ([(2,3)],4) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 2
[2,2] => ([(1,3),(2,3)],4) => ([],2) => ([(0,1)],2) => 1
[3,1] => ([(0,3),(1,3),(2,3)],4) => ([],1) => ([],1) => 1
[4] => ([],4) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 2
[1,4] => ([(3,4)],5) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[2,3] => ([(2,4),(3,4)],5) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 2
[3,2] => ([(1,4),(2,4),(3,4)],5) => ([],2) => ([(0,1)],2) => 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([],1) => ([],1) => 1
[5] => ([],5) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 4
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => ([(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 2
[1,5] => ([(4,5)],6) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[2,4] => ([(3,5),(4,5)],6) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,3] => ([(2,5),(3,5),(4,5)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 2
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([],1) => ([],1) => 1
[6] => ([],6) => ([],6) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 5
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 4
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 3
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(4,5),(4,6),(5,6)],7) => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 4
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => ([(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 2
[1,6] => ([(5,6)],7) => ([],6) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
[2,5] => ([(4,6),(5,6)],7) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[3,4] => ([(3,6),(4,6),(5,6)],7) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 3
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 2
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([],2) => ([(0,1)],2) => 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([],1) => ([],1) => 1
[7] => ([],7) => ([],7) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
delete endpoints
Description
Sends a graph to a maximal subgraph with no endpoints.
An endpoint of a graph is a vertex of degree one. Given an arbitrary graph, this map repeatedly searches for an endpoint and deletes it, until no endpoint remains. The result does not depend on the order of endpoints chosen, up to isomorphism. The map preserves the number of connected components. For a connected graph with at least one cycle, this map returns the 2-core.
Map
complement
Description
The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.